Understand contango and backwardation — Stock-Market notes

Overview

Contango and backwardation describe the relationship between futures prices and the current spot price (and between futures at different expiration dates). They reveal storage costs, financing costs, and convenience yields—critical for traders rolling futures contracts and understanding market structure.

Figure — Understand contango and backwardation

Core Concepts

Derivation: Cost of Carry Model

Question: Why does contango exist? Let's derive the relationship from arbitrage-free pricing.

Setup:

Current spot price: $S_0$
Futures price for delivery at time $T$ : $F_0$
Risk-free rate: $r$
Storage cost per unit time: $c$ (as % of spot price)
Convenience yield: $y$ (benefit of holding physical)

Arbitrage argument (cash-and-carry):

Consider two ways to own the asset at time $T$ :

Strategy A (buy forward): Enter a long futures at price $F_0$ today (costs nothing to enter). Set aside enough cash to pay $F_0$ at delivery: invest $F_0 e^{-rT}$ risk-free now, which grows to exactly $F_0$ at $T$ . Cost today: $F_0 e^{-rT}$ .
Strategy B (buy and carry): Buy the physical asset now for $S_0$ , pay storage over $[0,T]$ , and collect the convenience yield. The net present cost today of carrying the asset is $S_0$ plus the PV of storage minus the PV of convenience benefit, which (compounding continuously) is equivalent to a future obligation of $S_0 e^{(r+c-y)T}$ discounted back: cost today $= S_0 e^{(c-y)T}$ ... but crucially, both strategies deliver the same asset at $T$ .

No-arbitrage condition (both strategies must produce the same asset at $T$ for the same effective future cost): $F_0 = S_0 e^{(r+c-y)T}$

Why this step? If $F_0 > S_0 e^{(r+c-y)T}$ , an arbitrageur shorts the (expensive) futures, borrows $S_0$ to buy the physical, stores it (paying $c$ , earning $y$ ), and delivers into the short at $T$ . They repay the loan ( $S_0 e^{rT}$ ) plus net carry, and pocket the difference risk-free. This selling pressure pushes $F_0$ down until equality holds. If $F_0 < S_0 e^{(r+c-y)T}$ , the reverse trade (long futures, short physical) forces $F_0$ up.

Taking logs: $\ln(F_0) = \ln(S_0) + (r+c-y)T$

Interpretation:

Contango: $(r+c-y) > 0$ → $F_0 > S_0$ → futures price exceds spot, curve slopes up
Backwardation: $(r+c-y) < 0$ → $F_0 < S_0$ → futures price below spot, curve slopes down
The net cost of carry $(r+c-y)$ determines the curve shape

Worked Examples

Worked example Example 1: Gold Futures in Contango

Given:

Current gold spot: $S_0 =emptyset$ 2000/oz$
6-month risk-free rate: $r = 5\%$ p.a.
Storage + insurance: $c = 1\%$ p.a.
Convenience yield: $y = 0\%$ (gold has no industrial urgency)
Time: $T = 0.5$ years

Find: 6-month futures price $F_0$

Solution: $F_0 = S_0 e^{(r+c-y)T} = 2000 \times e^{(0.05 + 0.01 - 0)0.5}$ $= 2000 \times e^{0.03} = 2000 \times 1.0305$ $F_0 = \$2060.90$

Why this step? Gold is easy to store, no one desperately needs it immediately, so the futures price reflects financing + storage costs. The curve slopes upward (contango).

Check: $F_0 > S_0$ ✓ (contango confirmed)

Worked example Example 2: Crude Oil in Backwardation

Given:

Crude spot: $S_0 = \$ 80/barrel$
3-month risk-free rate: $r = 4\%$ p.a.
Storage cost: $c = 3\%$ p.a.
Convenience yield: $y = 10\%$ p.a. (refineries need oil NOW, supply tight)
Time: $T = 0.25$ years

Find: 3-month futures price $F_0$

Solution: $F_0 = S_0 e^{(r+c-y)T} = 80 \times e^{(0.04 + 0.03 - 0.10)0.25}$ $= 80 \times e^{(-0.03)(0.25)} = 80 \times e^{-0.0075}$ $= 80 \times 0.99253 = \$79.40$

Why this step? High convenience yield (refineries willing to pay premium for immediate oil) dominates storage + financing. Note the exponent is $(-0.03)(0.25) = -0.0075$ , giving $e^{-0.0075} \approx 0.99253$ . Futures are cheaper than spot (backwardation).

Check: $F_0 < S_0$ ✓ (backwardation confirmed)

Worked example Example 3: Rolling Futures in Contango (Roll Yield Loss)

Scenario: You hold a 1-month S&P 500 futures at $F_1 = 4500$ . Spot is $S_0 = 4480$ . As expiration nears, futures converge to spot. You roll to next month's contract at $F_2 = 4510$ .

Question: What's your roll yield?

Solution:

At expiration, $F_1 \to S_0 = 4480$ (convergence)
You close the expiring contract at $4480$ , open new position at $F_2 = 4510$
Roll cost: $4510 - 4480 = 30$ points (you buy the new contract 30 points higher than where the old one settled)
Roll yield: $\dfrac{4480 - 4510}{4510} = \dfrac{-30}{4510} = -0.665\%$ (negative = loss)

Why this step? In contango, you sell low (expiring contract converges down to spot) and buy high (next contract is elevated). Repeated rolling bleds returns—critical for long-term futures ETFs.

Backwardation scenario: If $F_2 = 4460$ , roll yield $= \dfrac{4480-4460}{4460} = +0.45\%$ (positive = gain). You sell high, buy low.

Common Mistakes

Common mistake Mistake 2: Ignoring Roll Yield in Long-Term Futures Holdings

Wrong idea: "I'll just keep rolling futures monthly; same as holding the asset."

Why it feels right: Each futures contract tracks the underlying, so rolling should preserve exposure.

The fix: In contango, each roll forces you to sell the expiring contract (converging down to spot) and buy the next contract (elevated above spot). This negative roll yield compounds over time, eroding returns even if spot doesn't move.

Calculation: If spot is flat but you lose 0.5% per monthly roll in contango, annual drag $\approx 1 - (0.995)^{12} = 5.8\%$ . Commodity ETFs (like USO for oil) can severely underperform spot due to persistent contango.

In backwardation: Roll yield is positive—you gain each roll, which can boost returns above spot performance.

Memory Aids

Recall Explain to a 12-Year-Old

Imagine ice cream trucks in summer (hot day = ice cream shortage).

Today's ice cream cone: $5 (everyone wants it NOW—hot outside!) **Promise to deliver ice cream in December**:$ 3 (winter, no one's desperate, trucks have plenty stock) This is backwardation—the "future delivery" is cheaper than "right now" because people are willing to pay extra for immediate ice cream when it's scarce.

Now flip it: It's winter. Ice cream today is $3. But storing ice cream until summer costs money (freezers, electricity). So a promise to deliver in summer costs$ 5—that's contango. The future price is higher than today's price because of storage + waiting costs.

In the stock market, contango = future contracts cost more than today's spot (like storing gold—you pay for safes, insurance). Backwardation = future contracts cost less than spot (like oil during a war—everyone wants oil TODAY, not in 6 months).

Connections

5.1.01-Futures-Contract-Basics – Foundation of futures mechanics
5.1.03-Futures-Pricing-and-Fair-Value – Costof-carry model underlying term structure
5.1.06-Rolling-Futures-Contracts – Practical impact of contango/backwardation on rolls
5.2.04-Commodity-Futures-and-ETFs – How contango destroys ETF returns
4.3.02-Implied-Volatility-Term-Structure – Similar concept in options (VIX futures often in contango)
6.1.05-SupplyDemand-Imbalances – Backwardation signals tight supply

Flashcards

What is contango?

A market condition where futures prices are above the current spot price and increase with later expiration dates, typically due to positive cost of carry (storage + financing > convenience yield).

What is backwardation?

A market condition where futures prices are below the current spot price and decrease with later expiration dates, typically due to high convenience yield from immediate demand or supply shortages.

Cost of carry formula for futures pricing?

F_0 = S_0 e^{(r+c-y)T}

where

r

= risk-free rate,

c

= storage cost,

y

= convenience yield,

T

= time to expiration.

When does contango occur (condition)?

When

r + c > y

(financing + storage costs exceed convenience yield), making futures more expensive than spot and far futures pricier than near ones.

When does backwardation occur (condition)?

When

y > r + c

(convenience yield exceeds carry costs), making futures cheaper than spot due to immediate demand/scarcity.

What is roll yield in contango?

The loss incurred when rolling futures in contango—you sell the expiring contract (converging down to spot) and buy the next contract (elevated above spot), creating negative returns even if spot is flat.

Why does backwardation signal tight supply?

High convenience yield reflects market participants willing to pay a premium for immediate delivery over waiting, indicating current scarcity or urgent demand.

How does contango affect commodity ETFs?

Persistent contango creates negative roll yield as ETFs repeatedly sell low (expiring contracts) and buy high (next month), causing the ETF to underperform the spot commodity over time.

What causes convenience yield to spike?

Supply disruptions, geopolitical events, or seasonal demand surges that make immediate possession of the physical asset more valuable than waiting for future delivery.

Futures convergence principle?

As expiration approaches, futures price converges to spot price (both must equal at delivery), regardless of whether the market is in contango or backwardation.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

**Cont

Test yourself — Futures

Connections